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NYU Calculus Class Uses Tangent Lines to Approximate Sine Function
1. Section 2.8
Linear Approximation and Differentials
V63.0121.002.2010Su, Calculus I
New York University
May 26, 2010
Announcements
Quiz 2 Thursday on Sections 1.5–2.5
No class Monday, May 31
Assignment 2 due Tuesday, June 1
. . . . . .
2. Announcements
Quiz 2 Thursday on
Sections 1.5–2.5
No class Monday, May 31
Assignment 2 due
Tuesday, June 1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 2 / 27
3. Objectives
Use tangent lines to make
linear approximations to a
function.
Given a function and a
point in the domain,
compute the
linearization of the
function at that point.
Use linearization to
approximate values of
functions
Given a function, compute
the differential of that
function
Use the differential
notation to estimate error
in linear approximations. . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 3 / 27
4. Outline
The linear approximation of a function near a point
Examples
Questions
Differentials
Using differentials to estimate error
Advanced Examples
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 4 / 27
5. The Big Idea
Question
Let f be differentiable at a. What linear function best approximates f
near a?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 5 / 27
6. The Big Idea
Question
Let f be differentiable at a. What linear function best approximates f
near a?
Answer
The tangent line, of course!
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 5 / 27
7. The Big Idea
Question
Let f be differentiable at a. What linear function best approximates f
near a?
Answer
The tangent line, of course!
Question
What is the equation for the line tangent to y = f(x) at (a, f(a))?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 5 / 27
8. The Big Idea
Question
Let f be differentiable at a. What linear function best approximates f
near a?
Answer
The tangent line, of course!
Question
What is the equation for the line tangent to y = f(x) at (a, f(a))?
Answer
L(x) = f(a) + f′ (a)(x − a)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 5 / 27
9. The tangent line is a linear approximation
y
.
L(x) = f(a) + f′ (a)(x − a)
is a decent approximation to f L
. (x) .
near a. f
.(x) .
f
.(a) .
.
x−a
. x
.
a
. x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 6 / 27
10. The tangent line is a linear approximation
y
.
L(x) = f(a) + f′ (a)(x − a)
is a decent approximation to f L
. (x) .
near a. f
.(x) .
How decent? The closer x is to
a, the better the approxmation f
.(a) .
.
x−a
L(x) is to f(x)
. x
.
a
. x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 6 / 27
11. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
12. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i)
If f(x) = sin x, then f(0) = 0
and f′ (0) = 1.
So the linear approximation
near 0 is L(x) = 0 + 1 · x = x.
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
13. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i) Solution (ii)
(π)
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3
and f′ (0) = 1. f′ π = .
3
So the linear approximation
near 0 is L(x) = 0 + 1 · x = x.
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
14. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i) Solution (ii)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = .
3
So the linear approximation
near 0 is L(x) = 0 + 1 · x = x.
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
15. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i) Solution (ii)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2
So the linear approximation
near 0 is L(x) = 0 + 1 · x = x.
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
16. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i) Solution (ii)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2
So the linear approximation
So L(x) =
near 0 is L(x) = 0 + 1 · x = x.
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
17. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i) Solution (ii)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2 √
So the linear approximation 3 1( π)
So L(x) = + x−
near 0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
18. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i) Solution (ii)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2 √
So the linear approximation 3 1( π)
So L(x) = + x−
near 0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus Thus
( ) ( )
61π 61π 61π
sin ≈ ≈ 1.06465 sin ≈
180 180 180
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
19. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i) Solution (ii)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2 √
So the linear approximation 3 1( π)
So L(x) = + x−
near 0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus Thus
( ) ( )
61π 61π 61π
sin ≈ ≈ 1.06465 sin ≈ 0.87475
180 180 180
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
20. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i) Solution (ii)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2 √
So the linear approximation 3 1( π)
So L(x) = + x−
near 0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus Thus
( ) ( )
61π 61π 61π
sin ≈ ≈ 1.06465 sin ≈ 0.87475
180 180 180
Calculator check: sin(61◦ ) ≈
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
21. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i) Solution (ii)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2 √
So the linear approximation 3 1( π)
So L(x) = + x−
near 0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus Thus
( ) ( )
61π 61π 61π
sin ≈ ≈ 1.06465 sin ≈ 0.87475
180 180 180
Calculator check: sin(61◦ ) ≈ 0.87462.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
22. Illustration
y
.
y
. = sin x
. x
.
. 1◦
6
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27
23. Illustration
y
.
y
. = L1 (x) = x
y
. = sin x
. x
.
0
. . 1◦
6
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27
24. Illustration
y
.
y
. = L1 (x) = x
b
. ig difference! y
. = sin x
. x
.
0
. . 1◦
6
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27
25. Illustration
y
.
y
. = L1 (x) = x
√ ( )
y
. = L2 (x) = 2
3
+ 1
2 x− π
3
y
. = sin x
.
. . x
.
0
. .
π/3 . 1◦
6
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27
26. Illustration
y
.
y
. = L1 (x) = x
√ ( )
y
. = L2 (x) = 2
3
+ 1
2 x− π
3
y
. = sin x
. . ery little difference!
v
. . x
.
0
. .
π/3 . 1◦
6
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27
27. Another Example
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 9 / 27
28. Another Example
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
Solution
√
The key step is to use a linear approximation to f(x) =
√ x near a = 9
to estimate f(10) = 10.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 9 / 27
29. Another Example
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
Solution
√
The key step is to use a linear approximation to f(x) =
√ x near a = 9
to estimate f(10) = 10.
√ √ d√
10 ≈ 9 + x (1)
dx x=9
1 19
=3+ (1) = ≈ 3.167
2·3 6
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 9 / 27
30. Another Example
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
Solution
√
The key step is to use a linear approximation to f(x) =
√ x near a = 9
to estimate f(10) = 10.
√ √ d√
10 ≈ 9 + x (1)
dx x=9
1 19
=3+ (1) = ≈ 3.167
2·3 6
( )2
19
Check: =
6
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 9 / 27
31. Another Example
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
Solution
√
The key step is to use a linear approximation to f(x) =
√ x near a = 9
to estimate f(10) = 10.
√ √ d√
10 ≈ 9 + x (1)
dx x=9
1 19
=3+ (1) = ≈ 3.167
2·3 6
( )2
19 361
Check: = .
6 36
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 9 / 27
32. Dividing without dividing?
Example
Suppose I have an irrational fear of division and need to estimate
577 ÷ 408. I write
577 1 1 1
= 1 + 169 = 1 + 169 × × .
408 408 4 102
1
But still I have to find .
102
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 10 / 27
33. Dividing without dividing?
Example
Suppose I have an irrational fear of division and need to estimate
577 ÷ 408. I write
577 1 1 1
= 1 + 169 = 1 + 169 × × .
408 408 4 102
1
But still I have to find .
102
Solution
1
Let f(x) = . We know f(100) and we want to estimate f(102).
x
1 1
f(102) ≈ f(100) + f′ (100)(2) = − (2) = 0.0098
100 1002
577
=⇒ ≈ 1.41405
408 . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 10 / 27
34. Questions
Example
Suppose we are traveling in a car and at noon our speed is 50 mi/hr.
How far will we have traveled by 2:00pm? by 3:00pm? By midnight?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 11 / 27
35. Answers
Example
Suppose we are traveling in a car and at noon our speed is 50 mi/hr.
How far will we have traveled by 2:00pm? by 3:00pm? By midnight?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 12 / 27
36. Answers
Example
Suppose we are traveling in a car and at noon our speed is 50 mi/hr.
How far will we have traveled by 2:00pm? by 3:00pm? By midnight?
Answer
100 mi
150 mi
600 mi (?) (Is it reasonable to assume 12 hours at the same
speed?)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 12 / 27
37. Questions
Example
Suppose we are traveling in a car and at noon our speed is 50 mi/hr.
How far will we have traveled by 2:00pm? by 3:00pm? By midnight?
Example
Suppose our factory makes MP3 players and the marginal cost is
currently $50/lot. How much will it cost to make 2 more lots? 3 more
lots? 12 more lots?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 13 / 27
38. Answers
Example
Suppose our factory makes MP3 players and the marginal cost is
currently $50/lot. How much will it cost to make 2 more lots? 3 more
lots? 12 more lots?
Answer
$100
$150
$600 (?)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 14 / 27
39. Questions
Example
Suppose we are traveling in a car and at noon our speed is 50 mi/hr.
How far will we have traveled by 2:00pm? by 3:00pm? By midnight?
Example
Suppose our factory makes MP3 players and the marginal cost is
currently $50/lot. How much will it cost to make 2 more lots? 3 more
lots? 12 more lots?
Example
Suppose a line goes through the point (x0 , y0 ) and has slope m. If the
point is moved horizontally by dx, while staying on the line, what is the
corresponding vertical movement?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 15 / 27
40. Answers
Example
Suppose a line goes through the point (x0 , y0 ) and has slope m. If the
point is moved horizontally by dx, while staying on the line, what is the
corresponding vertical movement?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 16 / 27
41. Answers
Example
Suppose a line goes through the point (x0 , y0 ) and has slope m. If the
point is moved horizontally by dx, while staying on the line, what is the
corresponding vertical movement?
Answer
The slope of the line is
rise
m=
run
We are given a “run” of dx, so the corresponding “rise” is m dx.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 16 / 27
42. Outline
The linear approximation of a function near a point
Examples
Questions
Differentials
Using differentials to estimate error
Advanced Examples
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 17 / 27
43. Differentials are another way to express derivatives
f(x + ∆x) − f(x) ≈ f′ (x) ∆x y
.
∆y dy
Rename ∆x = dx, so we can
write this as
.
∆y ≈ dy = f′ (x)dx. .
dy
.
∆y
And this looks a lot like the .
.
dx = ∆x
Leibniz-Newton identity
dy .
= f′ (x) x
.
dx x x
. . + ∆x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 18 / 27
44. Differentials are another way to express derivatives
f(x + ∆x) − f(x) ≈ f′ (x) ∆x y
.
∆y dy
Rename ∆x = dx, so we can
write this as
.
∆y ≈ dy = f′ (x)dx. .
dy
.
∆y
And this looks a lot like the .
.
dx = ∆x
Leibniz-Newton identity
dy .
= f′ (x) x
.
dx x x
. . + ∆x
Linear approximation means ∆y ≈ dy = f′ (x0 ) dx near x0 .
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 18 / 27
45. Using differentials to estimate error
y
.
If y = f(x), x0 and ∆x is known,
and an estimate of ∆y is
desired:
Approximate: ∆y ≈ dy .
Differentiate: dy = f′ (x) dx .
∆y
.
dy
Evaluate at x = x0 and .
.
dx = ∆x
dx = ∆x.
. x
.
x x
. . + ∆x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 19 / 27
46. Example
A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting
machine will cut a rectangle whose width is exactly half its length, but
the length is prone to errors. If the length is off by 1 in, how bad can the
area of the sheet be off by?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 20 / 27
47. Example
A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting
machine will cut a rectangle whose width is exactly half its length, but
the length is prone to errors. If the length is off by 1 in, how bad can the
area of the sheet be off by?
Solution
1 2
Write A(ℓ) = ℓ . We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in.
2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 20 / 27
48. Example
A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting
machine will cut a rectangle whose width is exactly half its length, but
the length is prone to errors. If the length is off by 1 in, how bad can the
area of the sheet be off by?
Solution
1 2
Write A(ℓ) = ℓ . We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in.
2 ( )
97 9409 9409
(I) A(ℓ + ∆ℓ) = A = So ∆A = − 32 ≈ 0.6701.
12 288 288
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 20 / 27
49. Example
A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting
machine will cut a rectangle whose width is exactly half its length, but
the length is prone to errors. If the length is off by 1 in, how bad can the
area of the sheet be off by?
Solution
1 2
Write A(ℓ) = ℓ . We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in.
2 ( )
97 9409 9409
(I) A(ℓ + ∆ℓ) = A = So ∆A = − 32 ≈ 0.6701.
12 288 288
dA
(II) = ℓ, so dA = ℓ dℓ, which should be a good estimate for ∆ℓ.
dℓ
When ℓ = 8 and dℓ = 12 , we have dA = 12 = 2 ≈ 0.667. So we
1 8
3
get estimates close to the hundredth of a square foot.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 20 / 27
50. Why?
Why use linear approximations dy when the actual difference ∆y is
known?
Linear approximation is quick and reliable. Finding ∆y exactly
depends on the function.
These examples are overly simple. See the “Advanced Examples”
later.
In real life, sometimes only f(a) and f′ (a) are known, and not the
general f(x).
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 21 / 27
51. Outline
The linear approximation of a function near a point
Examples
Questions
Differentials
Using differentials to estimate error
Advanced Examples
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 22 / 27
52. Gravitation
Pencils down!
Example
Drop a 1 kg ball off the roof of the Silver Center (50m high). We
usually say that a falling object feels a force F = −mg from gravity.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 23 / 27
53. Gravitation
Pencils down!
Example
Drop a 1 kg ball off the roof of the Silver Center (50m high). We
usually say that a falling object feels a force F = −mg from gravity.
In fact, the force felt is
GMm
F(r) = − ,
r2
where M is the mass of the earth and r is the distance from the
center of the earth to the object. G is a constant.
GMm
At r = re the force really is F(re ) = = −mg.
r2
e
What is the maximum error in replacing the actual force felt at the
top of the building F(re + ∆r) by the force felt at ground level
F(re )? The relative error? The percentage error? . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 23 / 27
54. Gravitation Solution
Solution
We wonder if ∆F = F(re + ∆r) − F(re ) is small.
Using a linear approximation,
dF GMm
∆F ≈ dF = dr = 2 3 dr
dr re re
( )
GMm dr ∆r
= 2
= 2mg
re re re
∆F ∆r
The relative error is ≈ −2
F re
re = 6378.1 km. If ∆r = 50 m,
∆F ∆r 50
≈ −2 = −2 = −1.56 × 10−5 = −0.00156%
F re 6378100
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 24 / 27
55. Systematic linear approximation
√ √
2 is irrational, but 9/4 is rational and 9/4 is close to 2.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 25 / 27
56. Systematic linear approximation
√ √
2 is irrational, but 9/4 is rational and 9/4 is close to 2. So
√ √ √ 1 17
2 = 9/4 − 1/4 ≈ 9/4 + (−1/4) =
2(3/2) 12
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 25 / 27
57. Systematic linear approximation
√ √
2 is irrational, but 9/4 is rational and 9/4 is close to 2. So
√ √ √ 1 17
2 = 9/4 − 1/4 ≈ 9/4 + (−1/4) =
2(3/2) 12
This is a better approximation since (17/12)2 = 289/144
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 25 / 27
58. Systematic linear approximation
√ √
2 is irrational, but 9/4 is rational and 9/4 is close to 2. So
√ √ √ 1 17
2 = 9/4 − 1/4 ≈ 9/4 + (−1/4) =
2(3/2) 12
This is a better approximation since (17/12)2 = 289/144
Do it again!
√ √ √ 1
2 = 289/144 − 1/144 ≈ 289/144 + (−1/144) = 577/408
2(17/12)
( )2
577 332, 929 1
Now = which is away from 2.
408 166, 464 166, 464
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 25 / 27
59. Illustration of the previous example
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
60. Illustration of the previous example
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
61. Illustration of the previous example
.
2
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
62. Illustration of the previous example
.
.
2
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
63. Illustration of the previous example
.
.
2
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
64. Illustration of the previous example
. 2, 17 )
( 12
. .
.
2
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
65. Illustration of the previous example
. 2, 17 )
( 12
. .
.
2
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
66. Illustration of the previous example
.
. 2, 17/12)
(
. . 4, 3)
(9 2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
67. Illustration of the previous example
.
. 2, 17/12)
(
.. ( . 9, 3)
(
)4 2
289 17
. 144 , 12
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
68. Illustration of the previous example
.
. 2, 17/12)
(
.. ( . 9, 3)
(
( 577 ) )4 2
. 2, 408 289 17
. 144 , 12
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
69. Summary
Linear approximation: If f is differentiable at a, the best linear
approximation to f near a is given by
Lf,a (x) = f(a) + f′ (a)(x − a)
Differentials: If f is differentiable at x, a good approximation to
∆y = f(x + ∆x) − f(x) is
dy dy
∆y ≈ dy = · dx = · ∆x
dx dx
Don’t buy plywood from me.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 27 / 27